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"... We present an algorithm for simplifying Fitch-style natural deduction proofs in classical first-order logic. We formalize Fitch-style natural deduction as a denotational proof language, N DL, with a rigorous syntax and semantics. Based on that formalization, we define an array of simplifying transfo ..."

We present an algorithm for simplifying Fitch-style natural deduction proofs in classical first-order logic. We formalize Fitch-style natural deduction as a denotational proof language, N DL, with a rigorous syntax and semantics. Based on that formalization, we define an array of simplifying transformations and show them to be terminating and to respect the formal semantics of the language. We also show that the transformations never increase the size or complexity of a deduction—in the worst case, they produce deductions of the same size and complexity as the original. We present several examples of proofs containing various types of superfluous “detours, ” and explain how our procedure eliminates them, resulting in smaller and cleaner deductions. All of the transformations are fully implemented in SML-NJ, and the complete code listing is available. 1.1

... fact it may well result in a dramatically (e.g., exponentially) larger proof, even when the original proof is fairly short and simple. (This has led some logicians to caution against cut elimination =-=[10]-=-.) By contrast, the result of our simplification procedure will never be larger than the original, and will indeed often be smaller and simpler. The remainder of this paper is structured as follows. T...

"... . In this paper we present a mechanism to define names for proof-witnesses of formulae and thus to use Gentzen's cut-rule in logic programming. We consider a program to be a set of logical formulae together with a list of such definitions. Occurrences of the defined names guide the proof-search ..."

. In this paper we present a mechanism to define names for proof-witnesses of formulae and thus to use Gentzen&apos;s cut-rule in logic programming. We consider a program to be a set of logical formulae together with a list of such definitions. Occurrences of the defined names guide the proof-search by indicating when an instance of the cut-rule should be attempted. By using the cut-rule there are proofs that can be made dramatically shorter. We explain how this idea of using the cut-rule can be applied to the logic of hereditary Harrop formulae. 1 Introduction The computation mechanisms both for logic and for functional programming are searches for cut-free proofs. First, in pure logic programming the achievement of a goal G w.r.t. a program P can be seen 1 as the search for a proof in Gentzen&apos;s intuitionistic sequent calculus LJ [Gen69], of the sequent P ) G, that by Gentzen&apos;s cut-elimination theorem can be cut-free [Bee89], [Mil90]; a -term found as a witness to a proof contains among...

"... We study the logic of dynamical systems, that is, logics and proof principles for properties of dynamical systems. Dynamical systems are mathematical models describing how the state of a system evolves over time. They are important for modeling and understanding many applications, including embedded ..."

We study the logic of dynamical systems, that is, logics and proof principles for properties of dynamical systems. Dynamical systems are mathematical models describing how the state of a system evolves over time. They are important for modeling and understanding many applications, including embedded systems and cyber-physical systems. In discrete dynamical systems, the state evolves in discrete steps, one step at a time, as described by a difference equation or discrete state transition relation. In continuous dynamical systems, the state evolves continuously along a function, typically described by a differential equation. Hybrid dynamical systems or hybrid systems combine both discrete and continuous dynamics. Distributed hybrid systems combine distributed systems with hybrid systems, i.e., they are multi-agent hybrid systems that interact through remote communication or physical interaction. Stochastic hybrid systems combine stochastic dynamics with hybrid systems. We survey dynamic logics for specifying and verifying properties for each of those classes of dynamical systems. A dynamic logic is a first-order modal logic with a pair of parametrized modal operators for each dynamical system to express necessary or possible properties of their transition behavior. Due to their full basis of first-order modal logic operators, dynamic logics can express a rich variety of system properties, including safety, controllability, reactivity, liveness, and quantified parametrized properties, even about

...equivalently modus ponens) can be very inefficient. For classical propositional logic, for instance, cut-based proofs can be exponentially smaller than the shortest corresponding cut-free proofs (see =-=[1]-=-). Still, the unrestricted use of the cut rule poses a serious challenge for proof-search. First proposed by Mondadori, KE tableaux for classical logic, thoroughly studied in [5, 8, 6], are a cut-base...

"... Algorithmic proof-search is an essential enabling technology throughout informatics. Proof-search is the proof-theoretic realization of the formulation of logic not as a theory of deduction but rather as a theory of reduction. Whilst deductive logics typically have a well-developed semantics of proo ..."

Algorithmic proof-search is an essential enabling technology throughout informatics. Proof-search is the proof-theoretic realization of the formulation of logic not as a theory of deduction but rather as a theory of reduction. Whilst deductive logics typically have a well-developed semantics of proofs, reductive logics are typically well-understood only operationally. Each deductive system can, typically, be read as a corresponding reductive system. We discuss some of the problems which must be addressed in order to provide a semantics of proof-searches of comparable value to the corresponding semantics of proofs. Just as the semantics of proofs is intimately related to the model theory of the underlying logic, so too should be the semantics of proof-searches. We discuss how to solve the problem of providing a semantics for proof-searches which adequately models both operational and logical aspects of the reductive system. 1

"... In [ 131 Parikh proved the first mathematical result about concrete consistency of contradictory theories. In [6] it is shown that the bounds of concrete consistency given by Parikh are optimal. This was proved by noting that very large numbers can be actually constructed through very short proofs, ..."

In [ 131 Parikh proved the first mathematical result about concrete consistency of contradictory theories. In [6] it is shown that the bounds of concrete consistency given by Parikh are optimal. This was proved by noting that very large numbers can be actually constructed through very short proofs, A more refined analysis of these short proofs reveals the presence of cyclic paths in their logical graphs, Indeed, in [6] it is shown that cycles need to exist for the proofs to be short. Here, we present a new sequent calculus for classical logic which is close to linear logic in spirit, enjoys cut-elimination, is acyclic and its proofs are just &amp;~errtar~ ~ larger than proofs in LK. The proofs in the new calculus can bc obtained by a srn~ll perturhntim of proofs in LK and they represent a geometrical alternative for studying structural properties of LK-proofs. They satisfy the constructive disjunction property and most important. simpler geometrical properties of their logical graphs. The geometrical counterpart to a cycle in LK is represented in the new setting by a spiwl which is passing through sets of formulas logically grouped together by the

"... This paper presents a measure of inference in classical and intuitionistic logics in the Gentzen-style sequent calculi. The measure for a proof of a sequent is the width of the proof tree, that is, the number of leaves of the proof tree. Then the measure for a sequent is the minimum value of the wid ..."

This paper presents a measure of inference in classical and intuitionistic logics in the Gentzen-style sequent calculi. The measure for a proof of a sequent is the width of the proof tree, that is, the number of leaves of the proof tree. Then the measure for a sequent is the minimum value of the widths of possible proofs of the sequent; if it is unprovable, the assigned value is +∞ � It counts the indispensable cases for possible proofs of a sequent. By this measure, we can separate between sequents easy to be proved and ones difficult; we can go further than provability and/or unprovability � It is motivated by some economics/game theory problem (bounded rationality). However, it would be not straightforward to obtain the exact value of this measure for a given sequent. In this paper, we will develop a method of calculating the value of the measure. We will apply our measure to various classes of problems, for example, to evaluate the difficulty of proving contradictory sequents. We also exemplify our measure with a problem of game theoretical decision making. 1.

... The inequality � Lw(�0) =� + ��� Lf(�0) =� × � shows a difference caused by (���)’s 6 . On the other hand, � Lf(�1) =� × ��� Lf(�2) =� + � assumes no (���)’s 6 This argument is reminiscent of Boolos =-=[3]-=-: In a first-order tableau system with equality and no (���)’s,hepresentedoneexamplewhereif(���)’s are additionally available, its poof became much smaller than the original proof. 21and is caused by...

...miniscent of the binary and the unary representation of numbers. This proof sequence is an exponential version of the sequences of Statman [36] and Orevkov [31] and has also been considered by Boolos =-=[6]-=-. In this paper we want to leverage this compression power of lemmas by automatically transforming cut-free proofs into proofs using compressing lemmas. 3. Proof-Theoretic Infrastructure Gentzen’s pro...

...ty. On the one hand, it is well known that proofs involving the cut rule (or, equivalently, modus ponens) can be dramatically shorter than the shortest cutfree proof of the same assertion (see, e.g., =-=[6]-=-, and the discussion in [14, Section 3.8], where the introduction of cut-based KE tableau systems is motivated). On the other hand it is obvious that unrestricted use of cuts may lead to infinitary br...